Property 7
\(\begin{aligned}\displaystyle\int f'(x)\cdot g(f(x))\,dx\\\Rightarrow\displaystyle\int g(u)\,du\end{aligned}\)
\(\begin{aligned}f(x)&=u\\f'(x)\,dx&=du\end{aligned}\)
To take the integral of the expression it could be first converted to \(g(u)du\)
\(\blacksquare\quad\displaystyle\int f'(x)\cdot f^n(x)\,dx=\samefrac{f^{n+1}(x)}{n+1}+c\qquad(n\ne-1)\)
\(\blacksquare\quad\displaystyle\int\samefrac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c\)
\(\blacksquare\quad\displaystyle\int f'(x)e^{f(x)}\,dx=e^{f(x)}+c\)
\(\blacksquare\quad\displaystyle\int f'(x)\cdot\sin(f(x))\,dx=-\cos(f(x))+c\)
\(\blacksquare\quad\displaystyle\int f'(x)\cos(f(x))\,dx=\sin(f(x))+c\)
1.
\(\displaystyle\int (x^2+1)^2\cdot2x\,dx=?\)
2.
\(\displaystyle\int (x^4+2)^6\cdot4x^3\,dx=?\)
3.
\(\displaystyle\int e^{\sin x}\cdot\cos x\,dx=?\)
4.
\(\displaystyle\int \samefrac{e^{2x}}{e^{2x}+4}\,dx=?\)
5.
\(\displaystyle\int \samefrac{\ln^3x}x\,dx=?\)
6.
\(\displaystyle\int 2\cdot\sin x\cdot\cos x\,dx=?\)
7.
\(\displaystyle\int \samefrac{2^{\sqrt x}}{2\sqrt x}\,dx=?\)
8.
\(\displaystyle\int \samefrac{\cos x}{\sin x}\,dx=?\)
9.
\(\displaystyle\int \tan x\,dx=?\)
10.
\(\displaystyle\int 5(x^3+2x)^4\cdot(3x^2+2)\,dx=?\)